The primary aim of this manuscript is to provide a mathematical framework for the geometric analysis of multi-scale discrete dynamical systems induced by maps. In essence, we aim to provide a discrete geometric singular perturbation theory, or simply (DGSPT), which parallels the established geometric singular perturbation theory (GSPT) for continuous-time fast-slow systems [1]}, [2]}, [3]}, [4]}, [5]}. Our starting point is motivated by the recently developed formulation in [5]} of GSPT for continuous-time, \(C^r-\) smooth fast-slow ODE systems in the general form
\(z^{\prime } = N(z) f(z) + \varepsilon G(z,\varepsilon ) ,\) 
